Problem 38 - Algebra and Trigonometry

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-14 x $$

Step-by-Step Solution

Answer

The constant that should be added to the binomial is 49, and the factored form of the resulting perfect square trinomial is \((x - 7)^{2}\)

Step 1: Identify the binomial's form

Looking at the given binomial, it is clear that it's in the form \(a^{2} - 2ab\), where \(a = x\) and \(2ab = 14x\). Hence, \(b = 14/2 = 7\).

Step 2: Calculate the constant to be added

The missing term in the perfect square trinomial is \(b^{2}\). So calculate \(b^{2}\) to get the constant that needs to be added to make the binomial a perfect square trinomial. Hence, \(b^{2} = 7^{2} = 49\).

Step 3: Write and factor the perfect square trinomial

Add 49 to the given binomial to form the perfect square trinomial: \(x^{2} - 14x + 49\). This trinomial factors into \((x - 7)^{2}\)

Key Concepts

Completing the Square

Completing the square is an algebraic technique used to solve quadratic equations, simplify expressions, and solve for variables in various equations. The goal is to transform a quadratic expression into a perfect square trinomial, which can be easily factored and then solved.

The method involves dividing the coefficient of the linear term (the term with the variable to the first power) by 2, squaring the result, and adding it to the constant term. For the given exercise, the binomial \( x^{2}-14x \) was transformed into a perfect square trinomial by adding 49, which is the square of \( 7 \), half of the coefficient 14. After this addition, the expression \( x^{2}-14x+49 \) became the perfect square trinomial \( (x-7)^2 \) that can be factored. This method is valuable for solving a variety of problems in algebra and calculus.

Factoring Trinomials

Factoring trinomials is the process of breaking down a cubic expression, typically a quadratic polynomial, into a product of binomials. A perfect square trinomial, a special kind of quadratic polynomial that resembles the square of a binomial expression, can be factored very cleanly as it is composed of two identical binomial factors. To factor trinomials, one must find two numbers that both add up to the middle term’s coefficient and multiply to the constant term.

With the exercise at hand, \( x^{2}-14x+49 \), since 49 is a perfect square and \( -14x \) is twice the product of 7 and \( x \), it indicates that the trinomial is a perfect square and can be immediately factored as \( (x-7)^2 \), demonstrating the case when identification of such patterns can simplify the process significantly.

Binomial Expressions

Binomial expressions consist of two terms that are either added or subtracted from one another. These algebraic expressions represent the simplest form of polynomials, next to monomials, and they frequently appear in the process of solving higher-degree polynomial equations.

In the context of the provided exercise, the original expression \( x^{2}-14x \) is a binomial. To turn it into a perfect square trinomial, a constant term is to be found and added, creating a scenario where the trinomial then represents the square of a binomial. Understanding the structure of binomials and the significance of their coefficients plays a crucial role in recognizing potential courses of action, such as completing the square.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (like \( x \), \( y \) or other letters), and arithmetic operations. These expressions may range from simple like binomials to more complex ones including higher degree polynomials. Algebraic expressions do not contain equality or inequality signs; those are reserved for equations or inequalities.

To solve algebraic expressions, particularly through operations like factoring or completing the square as with the exercise \( x^{2}-14x+49 \), students need to learn the rules of manipulating these expressions. This includes understanding how to apply the distributive property, combining like terms, and recognizing special products and factorizations, which are fundamental skills for successfully navigating algebraic problems.